Unbalanced and Minimal Point Equivalent Estimation Second-Order Split-Plot Designs

Abstract

Restricting the randomization of hard-to-change factors in industrial experiments is often required, resulting in a split-plot design structure. From an economic perspective, these designs minimize the experimental cost by reducing the number of resets of the hard-to-change factors. In this paper, unbalanced designs are considered for cases where the subplots are relatively expensive and the experimental apparatus accommodates an unequal number of runs per whole plot. We provide construction methods for unbalanced second-order split-plot designs that possess the equivalent estimation optimality property, providing best linear unbiased estimates of the parameters, independent of the variance components. Unbalanced versions of the central composite and Box–Behnken designs are developed. For cases where the subplot cost approaches the whole-plot cost, minimal point designs are proposed and illustrated with a split-plot Notz design.